If you are asking whether you can add more physical memory to an iPhone in the same way that you would a computer the answer is no. If you are looking to add additional file storage space the answer is "kinda". That involves using something like iCloud or DropBox to have sapce that the phone can access. There are some drives for IOS devices that provide more storage but they connect through the phone through Wifi.

As Sylv said, adding more physical storage to your iPhone is ... well I won't say impossible because I can't verify that but "not really feasible" should cover it.

You can use cloud-based storage to help free up space on your mobile device, but of course you'll need an internet connection to retrieve the files. For example, iTunes Match can be set to put your entire music collection (far more than you could ever store on your iPhone) in the cloud, making it all available to you.

There's a similar service for photos called EverPix (like iTunes Match, there's a modest annual fee). Flickr and SmugMug offer similar services as well.